Despite recent advances in the development of detailed plant radiative transfer models,
large-scale canopy models generally still rely on simplified one-dimensional (1-D) radiation models
based on assumptions of horizontal homogeneity, including dynamic ecosystem models, crop models,
and global circulation models. In an attempt to incorporate the effects of vegetation
heterogeneity or “clumping” within these simple models, an empirical clumping factor, commonly
denoted by the symbol

Solar radiation drives plant growth and function, and thus quantification of fluxes of absorbed
radiation is a critical component of describing a wide range of plant biophysical processes. Solar
radiation provides the energy for plants to carry out photosynthesis and drives the energy balance
and, thus, the temperature of plant organs

Rather than representing the absorption of radiation by individual leaves, radiation transport is
most commonly described statistically at the canopy level through the use of models. Using an
analogy to absorption of radiation due to a continuous particle-filled medium, classical radiation
transfer theory can be readily adapted to quantify radiation transport within a continuous medium of
vegetation as pioneered by

Perhaps one of the biggest challenges in the application of Beer's law is that it inherently assumes
that vegetation is homogeneous in space, but many, if not most, of the plant systems in which it is
applied are not homogeneous. For example, crops, savannas, coniferous forests, and even tropical
forests can have significant heterogeneity due to gaps that freely allow for radiation penetration
with near-zero probability of interception

An incredibly wide range of approaches of varying complexity have been used to develop radiation
transfer models applicable to heterogeneous canopies. The most robust and computationally expensive
approach is to explicitly resolve the most important scales of heterogeneity, such as with a
leaf-resolving model

Despite the wealth of available models for quantifying radiative transfer in heterogeneous canopies,
a critical knowledge gap still exists in which it is usually unclear which model is suited for a
particular application, and even a general sense of the errors associated with certain model
assumptions is often unknown. Models are commonly selected for historical reasons, based on ease of
implementation, availability of computational resources versus domain size, or presence of perceived errors
given the particular model assumptions. This uncertainty is driven by the fact that obtaining robust
validation data is exceptionally difficult, and often uncertainty in model inputs is comparable to
uncertainty in the model itself. Offsetting errors and coefficient “tuning” can lead to models
that perform exceptionally well in a particular case but may produce unacceptably large errors when
applied generally. These difficulties have led to a number of model intercomparison exercises in
which simulations are performed of synthetic or artificial canopy cases

This paper presents a critical re-evaluation of the theoretical basis of simplified one-dimensional (1-D) models of radiation transfer in heterogeneous canopies based on Beer's law. Due to the difficulties in objectively comparing and evaluating models based on field data, the performance of various models was explored by applying them in virtually generated canopies where the inputs are exactly known and comparing against the output of a detailed leaf-resolving model. The goal of the study was to better understand the implications of radiation model assumptions and uncertainty in model inputs in a wide range of canopy geometries in order to guide model selection in future applications.

The governing equation for radiation transfer in a participating medium is the radiative transfer
equation

If scattering and emission within the medium are neglected (i.e.,

As introduced previously, Eq. (

In a strict sense, each of these scales of heterogeneity violates the assumptions of Beer's law. One
obvious means of dealing with this heterogeneity is to use a more complicated model that explicitly
resolves the important scales of heterogeneity, such as a “multilayer” model that resolves
heterogeneity in the vertical direction

The issue of incorporating the effects of clumping in Beer's law models gained a heightened level of
attention in the early 1990s from investigators looking to use radiation measurements to invert
Beer's law for leaf area index (LAI) values

Within a few years, the clumping factor approach was incorporated into radiation transport models

Despite the seemingly attractive simplicity of the clumping factor approach, it has many
limitations. Foremost of these limitations is that in general the clumping factor

Hierarchical scales of spatial aggregation or clumping in a plant canopy.

Beer's law is only explicitly valid in a medium in which the probability of radiation interception
is homogeneous over some discrete scale. Thus, in order to apply Beer's law in heterogeneous
vegetation, we must segment the canopy into sections over which we can assume that the vegetative
elements are homogeneous in space. In a typical canopy, there may be multiple scales over which this
is applicable (Fig.

The cumulative probability of photon interception can be viewed as an aggregation of many “clumps”
consisting of a homogeneous medium of elements at a smaller scale. For each clump, the probability
of a photon intersection is assumed homogeneous in space, and thus the probability of interception
can be assumed constant. In this case, the cumulative probability of interception over all clumping
levels is

The product of Eq. (

The probability of a beam of radiation intersecting a single crown is the product of the probability
of intersecting the crown envelope and the probability of intersecting a leaf within the
envelope. At a solar zenith angle of zero, the probability of intersecting the crown envelope is
given by the ground cover fraction

In order to use the above expression to calculate the probability of interception for an entire
canopy of repeated crowns, we must assume a statistical distribution that describes the probability
of intersecting a crown envelope within a canopy. The two distributions that are commonly chosen are
the binomial distribution

If a Poisson distribution is assumed, the interception probability for the entire canopy is

This approach for estimating

To generalize this approach, we take the azimuthally symmetric plant spacing

Summary of inputs and equations needed to implement the BINOM model for canopies with spherical or cylindrical crowns.

Summary of 1-D models of radiation interception considered in this study.

As introduced above, the RTE (Eq.

A more robust approach for representing incoming diffuse radiation from the sky is to apply Beer's
law to any given direction in the sky and integrate across the upper hemisphere according to

For essentially all of the models considered in this work, the model input parameters are (1) the
leaf

Illustration of various crown envelope definitions applied to three different trees (two side view, one top view). The solid line is a sphere based on the vertical extent of the crown, the dashed line is a sphere based on the horizontal extent of the crown, and the dotted line is an ellipsoid based on the horizontal and vertical extent of the crown.

While it is possible to test the above modeling framework using field data, this approach is
severely limited by the lack of systematic variation in canopy architecture as well as experimental
errors that can become convoluted with model errors. As such, it can be difficult if not impossible
to use field data to rigorously evaluate and diagnose issues within models. In this work, an
alternative approach was used to evaluate model performance, which was based on the use of a
detailed 3-D leaf-resolving model to simulate radiation absorption in virtually generated
canopies. The advantage of this approach is that arbitrary canopy geometries can be generated in
which the exact geometry is known, which provides the necessary inputs for a 1-D model. The
limitation of course is that results are confined within the assumptions and accuracy inherent in
the chosen 3-D model. In order to minimize this limitation, the ray-tracing-based model of

A number of test cases were formulated to progressively test different aspects of each of the models
given in Table

Agreement between each of the 1-D models and the 3-D model were analyzed graphically and
quantitatively using the index of agreement

Visualization of virtual canopy geometries:

In order to isolate the effects of crown-scale clumping, a test case was considered in which the
canopy consisted of solid, opaque spheres of radius

To test generalization to geometries with anisotropic crowns, a case was considered with a canopy of
solid, opaque cylinders of radius

In order to test the combined effects of crown-scale clumping and leaf-scale attenuation, a test
case was considered in which the canopy consisted of spherical crowns containing homogeneous and
isotropic vegetation elements (Fig.

Similar to Case #3, an additional case was considered consisting of cylindrical crowns of radius

In order to test the models for more realistic canopy architectures, canopies of trees were
constructed using the procedural tree generator of

A potato plant canopy was generated to test the models in more realistic non-tree canopies
(Fig.

Summary of test case results. Agreement between the four models of radiation interception
is compared against the exact interception using the index of agreement (Eq.

Figure

For the east–west, row-oriented configuration (Fig.

Overall, the BINOM model performed equal to or better than the NIL99_B and NIL99_P models for every
canopy configuration. The lowest

Flux of intercepted radiation versus solar zenith angle for a half-diurnal cycle in a
canopy of solid spheres (see Fig.

Figure

Flux of intercepted radiation versus solar zenith angle for a half-diurnal cycle in a
canopy of solid cylinders (see Fig.

Figure

The OM_VAR model had very large errors as the canopy became increasingly sparse. For zenith angles
near 90

Flux of intercepted radiation versus solar zenith angle for a half-diurnal cycle in a
canopy of spherical crowns filled with uniformly distributed leaves (see
Fig.

Figure

The OM_VAR model had a very large overprediction of radiation interception at high zenith angles,
as in the spherical crown case. Unlike in the spherical crown case, the OM_CON model did not
perform well, particularly as the canopy became increasingly sparse. When crowns are cylindrical,
heterogeneity is no longer isotropic, especially as the plant spacing becomes large and as such
interception varies irregularly with

Flux of intercepted radiation versus solar zenith angle for a half-diurnal cycle in a
canopy of cylindrical crowns filled with uniformly distributed leaves (see
Fig.

Figure

Flux of intercepted radiation versus solar zenith angle for a half-diurnal cycle in a
canopy of trees (see Fig.

Figure

Flux of intercepted radiation versus solar zenith angle for a half-diurnal cycle in a
canopy non-tree plants (potato; see Fig.

The simplest models, based on an

The OM_VAR model

The assumption that the probability of intersecting a crown envelope follows a Poisson distribution
did not work well unless the canopy was very sparse. For most canopy densities, the Poisson models
significantly underpredicted the absorbed flux. It appeared that when the mean free path of
radiation propagation (which is related to the crown spacing) was not significantly smaller than the
actual radiation propagation distance through the canopy (which is related to the canopy height and
solar zenith), the assumption of a Poisson distribution was poor. The Poisson distribution assumes
that the canopy consists of a large number of “layers” of randomly positioned elements. When the
solar zenith angle is near vertical, the canopy consists of a single layer of crowns. For solid
spherical crowns, the probability of interception should be

The assumption that crown intersection followed a binomial distribution appeared to hold for all
canopy cases considered in this work. The binomial model predicts the correct interception at

The scope of the results of this study are clearly limited to cases of no scattering and no diffuse
radiation. These impacts were excluded from the study to focus on cases where, aside from
heterogeneity in the geometry, the assumptions of Beer's law should be exactly satisfied. Although
Beer's law is only valid along a single direction of radiation propagation, and its derivation
requires the removal of scattering terms in the RTE, variations have been derived that approximate
the effects of scattering and diffuse radiation within a 1-D model

It appears likely that many crop models, global ecosystem models, and land surface models overestimate radiation interception by applying the homogeneous Beer's law in heterogeneous
environments, which is sure to have important consequences for large-scale flux
estimates. Incorporation of the results of this work within these models is straightforward and
requires specification of either the ground cover fraction

Simplified models of radiation interception in heterogeneous canopies can be readily derived by
separating the canopy into hierarchical scales of clumping over which the probability of
interception can be assumed homogeneous in space over some discrete volume. The results of this work
demonstrated that very good predictions of whole-canopy interception can be achieved using simple
geometric models that consider only crown-scale and leaf-scale clumping (in the absence of
scattering). The probability of intersecting a plant crown was well represented by a (positive)
binomial distribution. This model calculates the probability of not intersecting a leaf within a
single crown and compounds this probability

For completeness, model equations taken from the literature are provided below as they were implemented and using the notation adopted in this paper.

Assuming that the probability of intersecting a crown envelope follows a (positive) binomial
distribution,

An obvious limitation of this approach is that the value of

Since

Helios code version 1.0.14 along with associated project files and output
files can be downloaded from the archived repository

BNB conceived the idea for the paper, performed simulations and analysis, and wrote the initial manuscript. ESK and MAP contributed to theoretical development and design of the study through discussions and editing of the paper.

The authors declare that they have no conflict of interest.

This research has been supported by the USDA NIFA (Hatch project no. 1013396) and the National Science Foundation, Directorate for Geosciences (grant no. 1664175). E. Scott Krayenhoff acknowledges funding from the Natural Sciences and Engineering Research Council of Canada.

This paper was edited by Gerd A. Folberth and reviewed by two anonymous referees.